(i) On the contrary, let us assume \(J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x)\ne \lbrace 0\rbrace \)  for a.e.
\(x \in X\) , then there exists a measurable set \(\Delta \subset X\)  such that \(\mu _{X}({\Delta })>0\)  and for all \(x\in \Delta \) , \(J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x)\ne \lbrace 0\rbrace \) . Let \(\lbrace d_1, d_2,\dots , d_n,\dots \rbrace \)  be a countable dense subset of \(\mathcal {H}\)  and \(U(x)=J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x)\) . Since \(U(x)\ne \lbrace 0\rbrace \)  on \(\Delta ,\)  then for each \(x\in \Delta \) , there exists \(d_n\)  such that \(P_{U(x)}d_n\ne 0\) . Let \(i(x)\)  be the least positive integer for which \(P_{U(x)d_{i(x)}}\ne 0\) . Consider the set \(\Delta _n=\lbrace x\in \Delta : i(x)=n\rbrace \) , then it is clear that \(\Delta =\oplus _{n \in \mathbb {N}}\Delta _n\) . Since \(\mu _{X}(\Delta )>0,\)  there exists \(N\in \mathbb {N}\)  such that \(\Delta _N\)  has positive measure. Define a function \(f\in L^2(X; \mathcal {H})\)  such that :
\(f(x)={\left\lbrace \begin{array}{ll}P_{U(x)}d_N, \ \mbox{when} \ x\in \Delta _N,\\0, \ \mbox{otherwise}.\end{array}\right.}\) 
Then using [1]}
, \(0\ne f\in S_{\mathcal {D}}(\mathcal {A})\cap S_{\mathcal {D}}(\mathcal {A}^{\prime })\) , a contradiction.
(ii) We prove that \(\sigma \big (S_{\mathcal {D}}(\mathcal {A})\cap S_{\mathcal {D}}(\mathcal {A}^{\prime })\big )=\lbrace x \in X: J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x)\ne \lbrace 0\rbrace \rbrace \)  as the later containment of sets is quite obvious. Let \(W:=\mathcal {S}_{\mathcal {D}}(\mathcal {A})\cap \mathcal {S}_{\mathcal {D}}(\mathcal {A}^{\prime })\) , \(E^{\prime }:=\mathcal {S}_{\mathcal {D}}(\mathcal {A})\ominus W\)  and \(F^{\prime }=\mathcal {S}_{\mathcal {D}}(\mathcal {A}^{\prime }) \ominus W\) , then \(W,E^{\prime }, F^{\prime }\)  are multiplication invariant spaces, and \(\mathcal {S}_{\mathcal {D}}(\mathcal {A})=E^{\prime }\oplus W\)  and \(\mathcal {S}_{\mathcal {D}}(\mathcal {A}^{\prime })=F^{\prime }\oplus W\) . Moreover, \( J_{\mathcal {A}}(x)=E^{\prime }(x)\oplus W(x)\)  and \(J_{\mathcal {A}^{\prime }}(x)=F^{\prime }(x)\oplus W(x)\) , by the pointwise projection property of \(\mathcal {D}\) -multiplication spaces. Now if \(x\in \sigma (\mathcal {S}_{\mathcal {D}}(\mathcal {A})\cap \mathcal {S}_{\mathcal {D}}(\mathcal {A}^{\prime }))=\sigma (W)\) , then \(\lbrace 0\rbrace \ne W(x)\subseteq J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x)\) .
Conversly, let \(x \in X\)  is such that \(J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x)\ne \lbrace 0\rbrace .\)  Then, there exists
\(0 \ne h \in J_{\mathcal {A}}(x)\cap J_{\mathcal {A}^{\prime }}(x).\)  Also there exists \(h_1 \in E^{\prime }(x), h_2 \in F^{\prime }(x)\)  and \(w_1, w_2 \in W(x)\)  such that \(h = h_1 + w_1 = h_2 + w_2.\)  Since \(h_1-h_2 \in W(x)^\perp \) 
and \(w_1-w_2 \in W(x)\)  implies \(h_1 =h_2\)  and \(w_1 =w_2.\)  Therefore, \(h = h_1 + w_1\)  with \(h_1 \in E^{\prime }(x) \cap F^{\prime }(x).\)  Also it is clear that, \(E^{\prime } \cap F^{\prime }= \lbrace 0\rbrace \)  which further by Proposition REF  implies \(E^{\prime }(x) \cap F^{\prime }(x) =\lbrace 0\rbrace .\)  Hence, \(h_1=0\)  and \(0 \ne h \in W(x) = \big (\mathcal {S}_{\mathcal {D}}(\mathcal {A})\cap \mathcal {S}_{\mathcal {D}}(\mathcal {A}^{\prime })\big )(x).\)
